 Now I want to talk today about the way the scaling relations arise between critical exponents and also give you a little bit of an introduction to the Ginsburg Landau theory of phase transitions and then go on from there. So let us first go back, take a look at what we deduced from the from mean field theory in the Ising class. We found that I did not calculate this but this exponent, the specific heat exponent is 0 in mean field theory. I did not compute this, it is a trivial calculation but I did not do it but it is alpha equal to 0 is the result in mean field theory. The magnetization exponent or order parameter exponent is a half, the susceptibility exponent is 1 and the critical isotherm exponent is 3. Remember that here m is h to the 1 third, so that was a relation here or m goes like h cubed, that was a relation sorry, h goes like m cubed was a relation. And then I introduced two other exponents, one was new for the correlation length and this was a half and this exponent was 0 essentially although again we did not talk about it in great detail and I said there are relations between these exponents here. Now the way they were arrived at originally was empirically, completely empirically and today we have an understanding of why these things are the way they are from mean field, from field, from field theoretic approach to it, from the renormalization group but in a sense it is ultimately experiment, it is ultimately basic experimental evidence. So through the 1960s people very patiently collated a lot of data and then finally we do not propose the following scaling function. Remember the equation of state we had for the Ising model and his problem, his point was the following, so it was followed shortly after that by other proposals for scaling more sophisticated proposals by other people such as Kadunov and so on, culminating in the renormalization group approach of Wilson and Fisher and others to calculate critical exponents in the function of dimensionality and the number of components of the order parameter. That is the sort of count jewel of equilibrium statistical mechanics but the idea of widown scaling was very simple. His point was that if you took the magnetization M in the critical region then this was equal to on the one hand 0 for t greater than 0, remember that t is t minus tc over tc. And but it was equal to minus t to the power beta apart from some constant functions etc in this fashion for t less than 0. So this was the magnetization and I should write this down here in the absence of a field. So one has a statement for M at t and h equal to 0. On the other hand if you looked at M at 0 and h for small h, oh let me let me let me mention here I should have done this that this is h the field divided by k Boltzmann. It is rescaled the magnetic field by k t and that I call little h, I did it implicitly but since you pointed out that there is a 1 over t let us call it little h that is the standard notation for it. Now you could ask what happens if you on the critical isotherm this stuff itself this goes like h to the power 1 over delta and delta was 3 in mean field theory. So is there a way to combine these 2 relations and get them out in one formula and then one expression and that is what widown did. What he did was to show that the huge pile of experimental data fitted the following scaling rule. The function M of t and h which would ordinarily be a function of 2 independent variables becomes now dependent only on one particular combination. So this thing here is equal to t to the power beta times something called a scaling function. So this f and this is for t positive so let us call it f plus h over t to the delta for t greater than 0 and it is minus t to the beta so it is just a dependence on the modulus some other scaling function of h over t to the delta minus t to the delta for t less than 0 and these functions f plus and minus are called scaling function. So his point was asymptotically close to the critical region where m is 0, h is 0, t is 0 asymptotically close all the data collapse onto this kind of functional form where this is some power, this is some power, this is some power etc. Now capital delta has nothing to do with little delta. He said there is a power delta such that you fit it in and you choose a proper delta and all the data collapse onto this sort of formula. The question is what is this capital delta etc. So delta is called the so called gap exponent but now let us see if it fits this how would you get that? You set f h equal to 0 and then you immediately see that now first of all it is clear that since m of t minus h equal to minus m of t h if you reverse the field the magnetization simply reverses at a fixed value of the temperature it is clear that both f plus or minus of whatever the argument is of u equal to minus f plus or minus whatever be the argument here the scaling functions of minus u. So the scaling function is an odd function in each case. Moreover above t c for t positive you get a 0 here right at h equal to 0. So it is guaranteed if you take f plus of 0 equal to 0 pardon me not necessarily that is the point not necessarily think about this it is not the odd character does not force it to be 0 because there could be a finite discontinuity right. I mean if you have a function that is odd then there are 3 possibilities if it is continuous then at 0 argument it is 0. If it is got an infinite discontinuity like 1 over x it is an odd function but it is got an infinite discontinuity that is perfectly alright but it could also have a finite discontinuity like the magnetization has right. So it could do this and indeed that is what is happening here that is an odd function but it is got a finite discontinuity and therefore does not vanish at the origin that is what the magnetization does is spontaneous magnetization okay. So what we require is that f minus of 0 is not equal to 0 some constant not equal to 0 then you are fine with this you get this immediately right. But the question is what is this delta for that one looks at the susceptibility. So since we know that chi t equal to delta m over delta h which is 1 over k Boltzmann t delta m over delta h at constant temperature and at h equal to 0 the 0 feels susceptibility. What you need to do is to differentiate this on both these functions here. This will give you the paramagnetic susceptibility and this gives you the paramagnetic susceptibility in the critical region just below then assuming that the slope of this functions at 0 argument is finite not 0 or infinity it immediately follows that the susceptibility chi t goes like t to the minus gamma which from here will go like t to the power beta minus delta because if I take out this function and differentiate it to first order in h since this fellow is 0 at 0 argument and I differentiate it you get a t to the delta in a denominator when I differentiate with respect to h so you get this relation here. So that immediately tells you that this gap exponent is not anything new but delta equal to beta plus gamma but now let us look at what happens if h is equal to 0 sorry if p equal to 0. So there you got to be a little more careful we want this result at little t equal to 0. So how is that going to happen? You want to make sure that on the critical isotherm at little t equal to 0 you get this cubic curve that is what you want and that will happen provided you see it can only happen because this fellow blows up at little t equal to 0 right. So the only way that can be saved is that there is a leading behavior here which is some power which vanishes when taken with this. So if f of u f plus or minus of u goes like u to some power as u tends to 0 then m of t h goes as t tends to 0 goes to t to the power beta and this is h to the lambda over t to the power delta lambda which is t to the power beta minus delta lambda h to the lambda and that is the same as h to the 1 over delta provided lambda equal to 1 over delta and beta is equal to lambda delta so that this goes away the t dependence cancels out that is only possibility but delta is beta plus lambda so it says therefore we have our second scaling relation our first one was that delta equal to beta plus gamma and now we discovered if I put in 1 over delta here beta plus gamma equal to beta delta so you get this extra relation therefore these are not independent exponents once you give me the others then this once you give me beta and gamma you note delta this is independent of mean field theory because this is extracted from experiment okay the actual numerical values of these are irrelevant the point is therefore empirically you will get numerical values for those but the point is across systems across different values of these exponents you still have these relations of course the scaling functions will change in different cases and so on that does not matter the point is that there is scaling at the critical point the sense that whenever you have function of several variables if you have a generalized homogenous function then you have what is called scaling because it means you can reduce the number of variables by 1 okay depending on how many combinations you can form you can reduce the number of independent variables and here you have reduced it from a function of t and h separately to apart from this power a single function of this combination okay now one can go on with this one can ask what happens in other in the case of other exponents for instance if you look at let us see what are the other exponents one can talk about if you look at the thing like a free energy the singular part because there is always a piece added like what I call phi naught earlier which is not singular is an uninteresting piece the free energy per unit volume I will it little f suitable free energy this guy scales like t to the power 2 minus alpha times again some scaling function let me just call it f once again offered h over t to the power delta in the critical region then the susceptibility sorry then m goes like delta f apart from the 1 over t factor etc over delta h okay and for small values of h near 0 this goes like t to the power 2 minus alpha minus delta times f of 0 sorry delta h times f prime of h whatever whatever that be and of course m will vanish in the critical region on the critical iso term and h is 0 that is perfectly okay but now the susceptibility chi t goes like this and there might be a minus sign here this goes like delta m over delta h apart from the 1 over t etc and this is now going to go like t to the power 2 minus alpha minus 2 delta times f double prime whatever so what does that give us part of the information we already know that says t minus 2 t minus 2 minus alpha minus delta is equal to beta because that is what the m should do and now chi t this thing should be gamma that is a susceptibility minus gamma 2 minus alpha minus 2 delta equal to minus gamma or you put beta plus gamma here it does not matter it is exactly the same thing so these two together will imply alpha plus twice beta plus gamma equal to 2 this is called the rush brook equality so we will add that to this alpha plus 2 beta plus gamma equal to 2 then you could ask what about the green function the correlation function itself because this was really where everything came from turns out that this g that we talked about as a function of r it is a vector r but we are looking at the dependence on this variable r t and h this is the correlation function suitably find course gained and integration etc etc done this guy has a scaling form 1 over r to the power d minus 2 plus 8 times again this I use the same symbol f for this f it is a function of 3 variables to start with but once you have this generalized scaling it is a function only of 2 combinations so it turns out to be a function of r t to the new and our old friend h over t to the delta again empirical evidence so from this exactly as we did before we can extract further relations among the exponents for instance and I will just write those results down you have a result which says 2 minus alpha equal to new d where d is the dimensionality remember dimensionality is appearing here in this place this is the only relation among all these for which the dimensionality is physically appearing it is called hyper scaling but it is a consequence of scaling finally and also you get gamma equal to new times 2 minus 8 now mean field exponents will not satisfy this because mean field theory is too only above the upper critical dimensionality which for the Ising class is 4 but if you substitute mean field exponents in 3 dimension then you get answers for alpha new etc which are independent of dimensionality if you put 0 here and you put a half here and 3 here clearly it is not satisfied but if you put 0 here and half here and 4 there then it is satisfied because that is the upper critical dimensionality all the others will be satisfied wherever dimensionality is involved mean field theory will flunk of course if you are below the upper critical dimensionality you have to go beyond it so the question is how do you do this how does one go beyond all this the answer which culminated in the renormalization group is rather long and tortuous but let me show you at least an indication of what the starting point is it is as follows we would first like to include fluctuations in the theory but before that we would like to find a systematic way of getting the relations we have such as the fact that for the order parameter M above TC there is one solution below TC there are two stable solutions and one unstable solution and so on we would like to get this from some principle which looks like a variational principle or some minimization of some energy functional here this energy functional is just a generalization of what I wrote down here for the Ising model made a little fancy what one does is instead of looking at individual magnetic moments one argues that in the critical region whole patches of linear dimension xi form where the systems get the system gets ordered and in the paramagnetic phase there are as many patches were down as up but as you get keeping as field in the positive direction as you get near the critical point these fellows grow at the expense of the other domains and finally the whole system in the thermodynamic limit the correlation length diverges and you have incipient magnet up okay. Now what one then does is to take this M and define a coarse grain magnetization for each patch of this size by writing M the size of the patches in linear dimension xi the correlation length of R you define this as 1 over the number of spins n xi at the center at the point R this is the center of it is the point R here summation i element of the patch or block s i so that is a coarse grain magnetization you define I am going to drop this subscript here just call it a feed and then you construct this land of free energy I should call it the functional of some kind which is geared so as to give me in equilibrium all the solutions that I had if I took the minimum of this function here so this is equal to an integral d d of R times you want a quadratic term and you want a quartic term so conventionally one writes this as a times t M squared of R this little t is t minus tc for reasons which we saw this function this coefficient has to change sign from positive to negative as you cross the critical point so you get this splitting of the minimum from a higher order minimum to 2 minima and a maximum in between by symmetry this we argued already in the absence of a field there is no linear term there are no odd powers of M by symmetry in the absence of a field plus the next term for stability one writes the B M to the 4 of R and if you put in a field you could put in now an inhomogeneous field it does not matter because it is still coupled the same way so minus M of R H of R put little h in d dimensions in arbitrary dimensions this is a positive number A that is a positive number B and it is temperature dependence is irrelevant on the critical region then the statement is the thermo dynamical equilibrium state is going to be obtained by taking the minimum of this L with respect to M we will soon convert it to at least briefly convert it to a field theory because what you really want to compute is the partition function which means you must have Z is e to the minus beta times this quantity here this is going to act like beta times this acts like some effective Hamiltonian if you like so you do e to the minus beta L and you have to trace but trace over what you have a continuous order parameter here so you trace over all configurations of this order parameter so it will be a path integral or a functional integral over all these guys but before that let us see how to get the equilibrium solutions so equilibrium solutions would imply delta L over sorry now I need a functional derivative this is now L is a functional of M of R and I want this functional derivative a functional is a function of a function and how do you do that well I presume you know the simplest rules for functional differentiation is this familiar or should I mention them can I go ahead and assume them yes is anyone who does not know what a functional derivative is so say it now the properties we need a very very simple we need to define well let us say let us not make a mistake of it suppose you have a functional of f of x function of a function this guy is a functional of M of R then the way you define delta f divided by delta little f of x in this fashion is to write this as f phi of y plus epsilon delta of x minus y minus f phi of y divided by epsilon and take the limit as epsilon tends to 0 and this guy is delta f so in particular this rule tells you that before we find this this rule tells you that the functional derivative of f of x or f of y with respect to f of x equal to delta of x minus y therefore if you have integral d y f of y and you do the derivative of that with respect to f of x here you take this in there get a delta function and the integral is 1 so this is equal to 1 that is the only rule you need to know so we can now differentiate this guy and what would you get let us see this here delta L over delta M of R equal to so let us make all these r primes r prime r prime r prime and start differentiating so I put it in here I get twice M of r prime okay and then I need the functional derivative of M of r prime with respect to M of r which is delta of r minus r prime I do the integral I get A t times M of r with the factor 2 so it is just like ordinary differentiation twice A t M of r plus 1 1 half B M squared of r minus and the equilibrium solution is given by setting this equal to 0 and you have to ensure that the second derivative is also got the right sign so it is at a minimum sorry this B M cube thanks right to one top so now in the absence of the field out here you are going to get your usual properties for a given sign of t now the point is that this is not enough you have neglect you have not taken into account fluctuations you need to have 2 things are going to happen if you have an inhomogeneity you could incidentally get stuck at some local minimum but thermal fluctuations will get you out of there to a global minimum but you need to put in a term which will take care of spatial variations now this is an unphysical model as it stands because it is exactly like saying that I have no connection between neighbouring patches at all it is too local it certainly costs a lot of energy by way of when you have a up patch with a down patch at the boundary at the domain mall you have a rapid change of the magnetization from up to down so it is going to cost you in gradient energy it is just like taking a string and when you vibrate the string there is a certain cost in energy if the string goes up and down too many times a gradient energy which is the reason why if you have a very short wavelength you have a higher frequency those are higher energy modes costs you more energy to excite those modes so the same kind of argument but how we do have to improve this functional here and how are we going to do that we need to take into account a term which takes not just m at one patch but m at the neighbouring patch as well so some gradient of m is going to appear and what is the simplest form that you can talk about well since it is going to be cannot be linear in m it has to be quadratic at the very least it has got to be symmetric and the m goes to minus m so it has got to be even powers of m and the simplest such term is a gradient squared so you add to this a half or again for differentiation purposes some coefficient gamma gradient of m of r oh gamma is an exponent right so let me call it I do not know what the standard notation for this is c for now because I am going to get rid of it I am going to scale out with this c in a second this is not the standard way in which this free energy is written I am going to scale it scale it out so what is this do to this equation here there you got to be a little careful you have to find this is exactly like the energy of a vibrating string except there is a nonlinearity here otherwise everything is linear as you can see so what is the functional derivative of this term if I differentiate it you are going to have twice so it is c over 2 times twice gradient of m itself times the functional derivative of the gradient so times delta so everything is r prime right r prime so let us write this out this term here if I take the functional derivative gives me d dr c over 2 into d dr prime twice gradient of m of r prime times delta over delta m of r gradient m of r prime but these two operators operations commute with each other so I can put this inside there and put a delta function so I get rid of this and write a gradient of delta of r minus r prime but you cannot directly handle the gradient of a delta function so you do an integration by parts and when you do that there is a surface term which vanishes in this case because there is a delta function sitting inside the bulk so with a minus sign this gradient acts on this so it is del dot del on m of r times a delta function so you have del squared of m of r prime times delta of r minus r prime you integrate you get del squared of m of r itself so this term becomes in the absence of a field you get an equation which has got an m cube term otherwise you would normally have a del squared plus constant times this guy which is like the Hemmholtz equation but you now have an extra m cube it is a non-linear equation and you put in an external field you have further complications there are many names for this equation but we are still not done because by the way it is when you do the renormalization group the scaling that you do to get rid of this whole thing is you divide through by c all the way through so you have at divided by c out here as a coefficient and you redefine your m by taking you want to make this thing here so you make you want to make this half gradient of whatever is squared so you set you want to find e to the minus beta l and write it as equal to e to the minus effective Hamiltonian so you got a beta c so you write square root of c beta times m of r equal to your order parameter field phi of r and then this h effective with identification of constants here let me write it in standard notation so h effective in phi of r squared plus the conventional notation as r naught phi squared plus 1 4 u naught phi 4 in the absence of a field this is like a phi 4 field theory so it starts at this point and then you can do dimensional analysis find out what are the dimensions of r naught u naught etc etc this guy is like 1 over length squared and so on so that is a piece of algebra I am not going to get into what I want to talk about instead is how does the system escape from a local minimum and fall into a global minimum that is how what the equilibrium state is because just setting this equal to 0 is not going to tell you whether it is a local minimum or a global minimum so you argue as follows you say that there is a time dependence in the problem and now you say you have m of r and t so an instantaneous configuration field configuration will change with time according to this equal to on the right hand side you say look the further away it is from a minimum the faster it relaxes so this relaxes with some coefficient let us call it gamma I use up little gamma for an exponent and what is left is delta L over delta m that is a single relaxation time kind of approximation you say there is a relaxation of this if this is 0 then of course this does not change at all but otherwise the deviation of this follow from the minimum value from 0 because that equilibrium this is 0 will tell you how fast this changes m changes okay. However you still have not allowed for random fluctuations because the thermal fluctuations are random so this by itself is like writing a Langevin equation with a friction term but without putting in the random force which you need for consistency so you put in now plus a zeta of r, t a noise white noise this is now a Langevin equation it is called the time dependent Ginsberg Landau equation but we know that such an equation we need to make some assumptions about this going to give us further information right so what are the assumptions we are going to make about this noise it is now a field it is now a configuration but a random field so we are going to assume that average zeta equal to 0 zeta of r, t correlation with r prime t prime equal to now some subtleties creep up here depending on whether the order parameter is conserved or not conserved and I am not going to get into that here now but in this problem this magnetization problem this thing is equal to some constant which is 2D times a delta function of r minus r prime delta of t minus t prime so it is really random noise this is the D dimensional to go further even that is not enough you have to do what you did in the Langevin equation you have to make some assumption about the nature of the probability distribution of this zeta here so you assume that it is Gaussian that the field configurations are distributed in a Gaussian management about some equilibrium value some fixed value of the configuration and then you have a Langevin equation but again you will discover there is a consistency condition you would have to satisfy between this and this and sure enough you would discover that D equal to 2 gamma k volts this is our old friend the fluctuation dissipation here but once you make a Gaussian approximate for this Gaussian probability distributions then we are all set we could write down a Fokker Planck equation for the probability distribution of the field configurations themselves and what would that probability distribution be well you would have to ask what is the probability of a configuration zeta of r prime t this is of course equal to what we mean by a probability distribution in any case is the expectation value of the the delta function of the random variable at any value it assumes so this is equal to the expectation value over what I will mention in a second of a delta function of zeta of what did I write m m we are looking at the field here we are looking at all this is gone this is phi sorry everything has been scaled out of phi is delta of phi of r t minus I need better notation delta function of this configuration minus phi okay this is the solution of the Langema equation for a given zeta noise and you want this to be equal to that solution so you put a delta function here and this is over the field configurations zeta so that is this p and it will satisfy a functional Fokker-Planck equation as you can as you can expect everything that you had as ordinary derivatives will become functional derivatives and so on I am not going to write that down just very makes a notation messy but I am just trying to motivate what happens here and then you look at the time scales you look at the critical slowing down dynamical indices and so on so this is where dynamic critical phenomena start at this point so it is a very straightforward extrapolation of whatever we did for a single particle we did the Langema equation we had certain basic ideas those ideas are just extrapolated to a field here a degree of freedom being replaced by continuous set field and we do not care how many spatial dimensions it has or how many components it has so the notation gets messier and messier but the basic intrinsic ideas are very straightforward so at this level it still be wrote write it down as a phenomenological thing but now it becomes serious business because the renormalization group analysis of this equation you start by identifying physical dimensions doing quote unquote dimension analysis on this identifying anomalous dimensions and then what happens you look at renormalization which is a fancy way of saying you try to scale things write things in a scale invariant manner and study the flow so to speak of this transformation and look at its fixed points so very in a nutshell that is really what critical point critical phenomenon analysis is all about but the time dependent part is also a straightforward extension of whatever we do most elementary cases so I said right in the beginning that the Langema equation would serve as a paradigm the kind of model for much more serious problems non-trivial problems and these are examples of such problems here we did not talk about hydrodynamics at all we didn't we just had a short excursion into kinetic theory but we didn't get to the hierarchy the BBG KY hierarchy and so on but I wanted to give a flavor for the way the subject is approached at least some aspects of it so I think I will stop here now